Higher Categories for the Working Mathematician

Session

Haugseng: An introduction to stable ∞-categories and spectra

27 Jul 2026, 15:30

Maximum (Georg-August-Universität Göttingen, Mathematisches Institut)

Conveners

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 1

• Rune Haugseng (NTNU Trondheim)

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 4

• There are no conveners in this block

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 3

• There are no conveners in this block

Haugseng: An introduction to stable ∞-categories and spectra: Lecture 2

• There are no conveners in this block

Description

This lecture series will give an introduction to spectra and their tensor product, focusing on categorical aspects of the ∞-category of spectra, as well as the broader context of stable ∞-categories. It will mainly be based on work of Lurie and Gepner–Groth–Nikolaus.

1. From abelian groups to additive ∞-categories. I will start by introducing the natural ∞-categorical analogues of commutative monoids and abelian groups, and then consider their connection to semiadditive and additive ∞-categories. We will also discuss the equivalence between commutative groups in spaces and infinite loop spaces.

2. Spectra and stable ∞-categories. We will motivate the ∞-category of spectra by considering several descriptions thereof: as non-connective delooping sequences, as cohomology theories, and as a natural enlargement of the stable homotopy category of finite CW-complexes. We will then define stable ∞-categories and consider spectra as an instance of stabilization.

3. The tensor product of spectra. We will first define symmetric monoidal ∞-categories and discuss some constructions of these. In particular, we will look at Day convolution, which gives us a way to define tensor products of commutative monoids and groups in spaces, as well as of spectra. We will then define algebras and modules in symmetric monoidal ∞-categories, which in particular lets us consider ring spectra and modules over them.

4. The presentable tensor product and the universal property of spectra. In the last lecture we will briefly introduce presentable ∞-categories and their tensor product. Then we will explain how presentable stable ∞-categories can be described as being precisely modules over the ∞-category of spectra, and how this gives a universal property of the symmetric monoidal ∞-category of spectra, as well as the analogous results for presentable (semi)additive ∞-categories.

1. Lecture

Rune Haugseng (NTNU Trondheim)

27/07/2026, 15:30